The Inverse Hyperbolic Sine Function

The graph of the hyperbolic sine
function y = sinhx is sketched in Fig. 1.1. Clearly sinh
is one-to-one, and so has an
inverse, denoted sinh–1. The inverse hyperbolic sine
functionsinh–1 is defined as follows:

The graph of y = sinh–1x is the mirror image of that of y = sinhx
in the line y = x.
It's shown in Fig. 1.1. We have
dom(sinh–1)
= R and range(sinh–1) = R.

Fig. 1.1

Graph of y = sinh–1x.

The Inverse Hyperbolic Cosine Function

Fig. 1.2

Graph of y = cosh–1x.

The Inverse Hyperbolic Tangent Function

The graph of the hyperbolic tangent function y = tanhx
is sketched in Fig. 1.3. Clearly tanh is one-to-one, and so
has an
inverse, denoted tanh–1. The inverse hyperbolic tangent
functiontanh–1 is defined as follows:

Fig. 1.3

Graph of y = tanh–1x.

The Inverse Hyperbolic Cotangent Function

The graph of the hyperbolic cotangent function y = cothx
is sketched in Fig. 1.4. Clearly coth is one-to-one, and
thus has
an inverse, denoted coth–1. The inverse hyperbolic cotangent
functioncoth–1 is defined as follows:

Fig. 1.4

Graph of y = coth–1x.

The Inverse Hyperbolic Secant Function

Fig. 1.5

Graph of y = sech–1x.

The Inverse Hyperbolic Cosecant Function

The graph of the hyperbolic cosecant function y = cschx
is sketched in Fig. 1.6. Clearly csch is one-to-one, and so
has
an inverse, denoted csch–1. The inverse hyperbolic cosecant
functioncsch–1 is defined as follows:

Fig. 1.6

Graph of y = csch–1x.

Example 1.1

Prove the identity:

Note

Recall that the inverse of the natural exponential function
is the natural logarithm function. Since the hyperbolic functions
are defined in terms of the natural exponential function, it's not surprising
that their inverses can be expressed in terms
of the natural logarithm function. Also see Problem
& Solution 1 and Problem & Solution 2.